ASYMPTOTIC THEORY OF MULTIDIMENSIONAL CHAOS

A delay-differential equation epsilonu(t) + u(t) = f(u(t-1)), 0 less-than-or-equal-to t < infinity, and its generalization are investigated in the limit epsilon --> 0, when the attractor's dimension increases infinitely. It is shown that a number of statistical characteristics are asymptotically independent of epsilon. As for the attractor, it can be regarded as a direct product of O(1/epsilon) equivalent "subattractors," their statistical characteristics being asymptotically independent of epsilon. The results enable one to predict some characteristics of the attractor with fractal dimension D much greater than 1 for the case epsilon much less than 1, when they are inaccessible numerically. The approach developed seems to be applicable for a wide class of spatiotemporal systems.